Insight Problems About Pendulums.

Sagredo: You give me frequent occasion to admire the wealth and profusion of nature when, from such common and even trivial phenomena, you derive facts which are not only striking and new but which are often far removed from what we would have imagined. —Galileo Galilei. Dialogues Concerning Two New Sciences. Translated by Henry Crew and Alfonso de Salvio. 1914, Dover, 1954.

Here's a question for you folks who like questions that require insight and understanding:

A 1kg laboratory weight on a weight hanger is suspended as a pendulum, with a 1 meter length suspension. Its period for small angle oscillations is determined. Now another 1kg weight is placed on top of the other one, and the period is re determined. The period with both weights is

a) much greater than before,
b) slightly greater than before,
c) unchanged,
d) slightly less than before,
e) much less than before.

This question was first posed to me by James Randi in another context, a context I'll reveal later.

Answer and discussion.
The period decreases slightly when the second weight is added. The sophisticated reason is that while an increase of mass alone doesn't change the period, the placement of the additional mass above the other one shortens the radius of gyration. But this problem doesn't require knowledge of the definition of radius of gyration, nor any mathematical analysis, knowledge of physical pendulums, nor even the equation for the period of a simple pendulum.

This question is suitable for people innocent of physical pendulums and their theory. It assumes only observational familiarity with swinging suspended bodies, of the sort you'd find in a high school physics text. The outcome is definitely counter intuitive. And so Galileo is honored as the first to tell us that the period of a pendulum is independent of the amplitude, to a first approximation (also counter intuitive) and that the period is independent of the mass. The beauty of the pendulum is that it is so counter intuitive in so many ways, which teaches students not to rely on naive intuition, but to develop more correct and powerful intuition from experimentation and study. Too many students think that they are born with correct intuitive knowledge (a mistake also made by the ancient Greek thinkers).

This gets at the kind of reasoning Galileo was famous for. We reason from a minimum amount of knowledge about the behavior of pendulums to arrive at predictions for the behavior of pendulums of new configuration.

We may assume that the student is already convinced that the period of a simple pendulum is independent of the mass, but very much dependent on the distance of that mass from the fulcrum, and that a simple pendulum with shorter suspension has shorter period. That's all the prior information needed. This of course, assumes that the pendulum bob is small in size compared to the suspension length, and may be treated as a point mass.

But we are stalking bigger game. We'd like to learn something of pendulums with larger bobs, where the point-mass approximation is clearly not appropriate.

Figure 1. A coin swinging as a pendulum.

Figure 2. Superposing two swinging coins.

First let's inquire why the period of a pendulum is independent of the mass. Galileo might have argued as follows:

Imagine such a mass swinging. Say a thin coin, suspended from its edge by two strings (Fig. 1). Now consider another identical coin suspended, and swinging in the same manner. They swing in synchronism (Fig. 2A). One may move these two pendulums closer together. Consider moving them very close together, perhaps even touching. They swing together, touching all the time (small amplitude is assumed). The period is still the same (Fig. 2B). It would be no different if they were glued together creating one mass twice as much as the first. The period would still be the same. We conclude that the period is independent of the mass.

Well, perhaps the periods could be just a little different. Gluing the coins together would prevent the small up/down relative motion of the two coins, perhaps altering the result just a little bit. But those are just the corrections the high powered physical pendulum analysis gives when you deal with an extended object rather than a point mass! And these differences surely decrease as the coins are made thinner, and in the limit, go to zero as the thickness of the coins goes to zero.

Figure 3. Superposing two pendulums.

So, accepting that adding mass at the same distance from the fulcrum doesn't change the period we now consider the effect of moving the mass with respect to the fulcrum. Start with a mass on a light, but rigid rod (Fig. 3A). Moving the mass up the rod will result in a shorter period of oscillation (Fig. 3B). We take this result from experimental observation); one has to start with some knowledge of the real world! Now superpose the two cases by placing two masses on the same suspension rod, but at different positions (Fig 3C). This system would be forced to have the same period, but that period would be less than the natural period of the one with the greater distance, and larger than the natural period of the one with the shorter distance. I.e., a period in between the natural periods of the two masses. And in our problem, that's a period shorter than the original period. This analysis is not quantitative, but the question didn't require a quantitative answer.

Now to the real world fact which inspired this. Large tower clocks have huge weights on long suspensions. They are often adjusted by adding small weights on top of the main weight. Sometimes a tray or pan is provided there for that purpose. Adding weight shortens the period. The clockwork which rings the bell "Big Ben" is adjusted this way, using the old English Pennies as weights (they are copper coins the same size as the U.S. half dollar). This method is apparently common practice in Europe. For an example, I excerpt below an item found on the net. Dr. King can be excused for being cavalier about the distinction between "center of mass" and "distance to the center of gyration", since with these huge clocks with long suspensions you'd be hard put to measure the difference between the two.

The Cambridge Chimes.

In 1793, a new clock by Thwaites of London was installed...

These much admired chimes were copied in 1859 for the new clock and bells in the Palace of Westminster (commonly collectively known as ``Big Ben'', which is in fact the name of the hour bell alone)...

Regulation of the clock.

The following information was provided by the University clock keeper, Dr Frank King.

The clock at Great S. Mary's is usually very reliable and gives the time to within a few seconds of legal time (G.M.T. or B.S.T.).

The clock is normally checked weekly at the first stroke of the hour bell at 8 a.m. each Sunday and the error relative to legal time is recorded. The weekly records date back many years and show that the standard deviation of the error is around 2 seconds.

The pendulum is equipped with an annular tray which is used for holding coins, typically about 10 2p pieces. After each weekly observation an appropriate adjustment is made to the number of coins on the tray according to the following formula:

Adjustment = (desired gain + last week's gain) / 1.3

The gain is measured in seconds and the result, rounded to the nearest half integer, gives the amount in pence to be added to the tray.

For example if the clock is 2 seconds slow and lost 1.5 seconds the previous week then the desired gain is 2 seconds and last week's gain is 1.5 (a loss is a negative gain). The formula suggests adding (2+1.5)/1.3 = 2.69 p to the tray. On rounding to the nearest half-integer, this result indicates that 2.5p should be added to the tray. A ha'penny is kept by the clock.

The position of the hands cannot be read to satisfactory precision which is why the time of the first stroke of the hour is used as a reference. Moreover, the error in the time of the first stroke is slightly different at different hours of the day which is why a particular hour (8 a.m. in practice) needs to be used for observational consistency.

The period of a pendulum is governed by the formula

[6]

T = 2π √(L/g)

where L is the length of the suspension. The annular tray is above the pendulum bob so if you add pennies to the tray you raise the centre of gravity and hence shorten the simple equivalent length marginally. This reduces the period and so speeds up the clock.

The pendulum incorporates a temperature compensation mechanism but it suffers from time lag and requires human intervention to help it on its way.

It is worth noting that the time is almost never adjusted, but only the rate.

The clock is stopped for occasional maintenance and when the clocks go back an hour in October each year. It may take a week or two for the timing to recover after such incidents which typically occur once or twice a year.

The practicality of the method is obvious. The correction may be made "on the fly" without stopping the swing of the pendulum. Grandfather clocks are adjusted by a screw which moves the bob up and down the pendulum shaft. This requires stopping the clock during adjustment.

Another pendulum problem:

Suspend a flat square metal plate from strings at its four corners, so that it lies in a horizontal plane and the strings are parallel and vertical. Let the string lengths be, say 1.5 times the length of an edge of the plate. Pull the plate aside and it swings like a pendulum, remaining horizontal at all times. Find its period of oscillation. Now cut two strings which support one edge. The plate drops and now hangs from two parallel strings in a vertical plane. Pull the plate aside and release it so it swings in its own plane, its upper and lower edge remaining horizontal at all times, and its other two edges remaining vertical.

Clearly the average distance of the plate's mass from the suspension point is significantly greater than before, but its mass hasn't changed. Is its period now

a) very much greater than before,
b) slightly greater than before,
c) the same as before,
d) slightly smaller than before,
e) very much smaller than before?

The outcome is easily tested by timing, say, 20 swings with a stopwatch. Or your pulse, as Galileo might have done. Every physics student should be required to read Galileo's Dialogues Concerning Two New Sciences.

This is easy to demonstrate. Find or construct a rigid support to suspend strings from. A flat metal plate, the larger the better for class use. For smaller groups I've used a large flat Meccano or Erector metal plate, about 5 x 7 inches.

Note that the direction of swing is such that in the second case (B) the plate moves in its own plane. Compare the two periods in the two cases by timing about 20 swings. I've done this for audiences of physics teachers, many of whom were surprised at the result.

In my demos for teachers lecture I remark "If we teachers can't understand simple everyday things like this well enough to predict them, and explain them to students, we kid ourselves if we suppose we can teach subtle things like quantum mechanics." As you know I like to be provocative.

Then I slide into a consideration of the mechanism of the platform rocker chair. Don't teachers claim they like physics applications to everyday things?

Analysis of the flat plate pendulums.

Figure 5. A horizontal rod pendulum.

We need to compare a swinging plate with what we have learned from the previous analysis. Start with a small coin as before (Fig. 1). We have shown that we can combine two such coins into a thicker coin swinging with the same period (Fig. 2). We can continue this process to build up a swinging rod (Fig. 5A). The period is still the same. The rod swings in a plane, and all its positions are mutually parallel.

Now imagine a second identical rod, with the same radius of suspension. It's period is the same as the first rod. The second one can be positioned just above, and very close to the first one if its suspension point is raised one rod diameter. We modify the suspension so that the suspension strings can't interfere with the motion of either rod, but we keep the suspension length the same as before, and the same for both rods. The periods are still the same, and the rods can swing with very little relative motion. They could be glued together and the strings to the bottom one cut, and they would still have the same period, a period determined by the suspension length to the fixed supports.

In this way we can add horizontal rods, fastening them together to form a plate of any size. It swings as shown in Fig. 6A, with period solely determined by the length of the suspension strings. It will have the same period as the rod of Fig. 6B.

The plate swings in an interesting manner, each edge moves so that it is always parallel to a fixed axis.

Independence of pendulum period on mass.

Figure 6. Flat plate pendulum and rod pendulum.

Is there an intuitive or conceptual argument that the pendulum period should not depend on the mass of the bob? Of course, or I wouldn't mention the question here.

Increased mass increases mg and the string tension in proportion, thereby increasing the restoring force f in the same proportion. For simple harmonic motion, the restoring force must be proportional to the displacement. Changing the mass doesn't change that proportionality. The acceleration of the bob is given by Fx = ma. Since Fx is proportional to m, the acceleration isn't dependent on the mass.

The language we use.

The pendulum formula ought to be written:

[1]

T = 2π √(R/g)

where R is the radius of the bob's motion. Using L for the radius makes students think that the length of a physical suspension cord and its point of attachment have something to do with the behavior of the pendulum. In the case of the pendulum on the curved ramp, that's obviously not true.

But what about inertia?

If the mass of the bob has no effect on the period, does that mean the inertia of the bob has no effect? Not so fast! Inertia has two forms, linear and angular. Mass is the measure only of linear inertia.

A mass sliding without friction on the curved track has the same period as the pendulum. But what about a ball rolling on the same curved track? The rolling ball has "extra" inertia due to rotation which is not affecting the restoring force.

Test your understanding.

(1) The method used above is often called "Galileo's principle of superposition." It says that when two physical influences on a process are independent (do not change each other) then when both influences act simultaneously, the combined influence on a process is the sum of the influences of each alone. (The word "sum" is a bit tricky here, for in some processes it might be an algebraic sum, but in others a vector sum is required.) Galileo used this principle in two notable examples:

(A) When a body falls a certain height it accelerates straight downward. When a projectile is fired horizontally, it moves with constant speed unless affected by another force. When the projectile is fired horizontally and gravity also acts on it, its motion is the combination of the downward accelerated fall and the horizontal constant speed motion. These two motions add as vectors. Develop this argument as we did above with the pendulums.

(B) When two bodies of different size and mass fall from the same height, they take very nearly the same time to cover a given distance. I.e., a body's fall isn't affected much by its mass, and in a vacuum, isn't affected at all by its mass. Develop this argument as we did above with the pendulums.

(2) Can you think of some everyday examples of processes for which the superposition principle would not work?

Rolling Pendulum Problems.

How often thinking about physics leads back to pendulums. You could make a respectable course in classical mechanics where every example and exam question involves a pendulum.

Problem 1: A small four-wheeled miniature car rolls on a curved circular track. The wheels have only a small mass relative to the cart, and we can ignore them for now. The track and the car's motion are in a vertical plane, like a section of a roller coaster. You expect the cart will execute motion back and forth, obeying the formula for the period of a simple pendulum. Now if a spherical ball the same mass as the cart rolls on the same track, would its rotation matter? After all, the rotation gives the ball additional rotational inertia. So would the ball have a different period than the car? If so, has it a larger period, or smaller? Insightful answers get extra credit.

Problem 2: What about using a larger and heavier ball on the same track? Will it have the same period as a smaller one? Their periods can be measured with a stopwatch.

Problem 3: Race a large ball and a smaller one on the same track. Start them simultaneously, perhaps an inch apart. When they return to the starting end, will they still be an inch apart? See, you don't need the stopwatch.

These are easy to test. A section of right angle aluminum stock 4 feet long or more doesn't cost much. Affix it to a solid table with clamps at either end, and put compression on it so it bends into a shallow curve. Slight curve is best, for longer periods of motion. Obtain several balls of different size and mass, and make a small cart from Meccano with four wheels that will smoothly roll on the track. Add weights to it so it matches the weight of one of the balls.

Answer: They will all have the same period. Just as with a simple pendulum, the period doesn't depend on the mass (inertia), and these experiments show it doesn't depend on the rotational inertia, either. Nor does a small amount of friction matter, for period doesn't depend on amplitude, and the friction only decreases the amplitude. If any differences in the periods are seen in the experiments, that is likely due to other factors.

—Donald E. Simanek, 2005, 2009, 2112

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